Discrete derivatives of permutations are characterized by consecutive partial-sum sets, and two-valued derivatives are exactly coprime pairs (a,-b), with explicit cyclic constructions.
Drakakis, An introduction to Costas arrays: http://www1.spms.ntu.edu.sg/∼ccrg/documents/basicTalkSingapore.pdf
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Permutation Matrices, Their Discrete Derivatives and Extremal Properties
Discrete derivatives of permutations are characterized by consecutive partial-sum sets, and two-valued derivatives are exactly coprime pairs (a,-b), with explicit cyclic constructions.